非Lipschitz条件下高维McKean-Vlasov随机微分方程解的存在唯一性

马丽; 孙芳芳

doi:10.21656/1000-0887.440010

非Lipschitz条件下高维McKean-Vlasov随机微分方程解的存在唯一性

doi: 10.21656/1000-0887.440010

马丽^{1, 2,},
孙芳芳^1, ,

1.
海南师范大学数学与统计学院，海口 571158
2.
海南师范大学数据科学与智慧教育教育部重点实验室，海口 571158

基金项目:

国家自然科学基金(地区科学基金)项目 11861029

海南省高层次人才项目 120RC589

详细信息

作者简介:
马丽(1979—), 女, 副教授, 博士(E-mail: malihnsd@163.com)

通讯作者:
孙芳芳(1998—), 女, 硕士生(通讯作者. E-mail: sunff1207@163.com)

中图分类号: O211.63
计量
- 文章访问数: 429
- HTML全文浏览量: 192
- PDF下载量: 44
- 被引次数: 0
出版历程
- 收稿日期: 2023-01-10
- 修回日期: 2023-03-11
- 刊出日期: 2023-10-31

Existence and Uniqueness of the Solutions to High-Dimensional McKean-Vlasov SDEs Under Non-Lipschitz Conditions

MA Li^{1, 2
,},
SUN Fangfang^{1
, ,}

1.
School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, P.R.China
2.
Key Laboratory of Data Science and Smart Education, Ministry of Education, Hainan Normal University, Haikou 571158, P.R.China

摘要

摘要: 研究了一类漂移系数不连续的高维McKean-Vlasov随机微分方程及相应的粒子系统解的存在唯一性. 在漂移系数关于空间变量逐段Lipschitz连续的条件下，首先利用Zvonkin变换将方程转换为漂移系数为Lipschitz连续的McKean-Vlasov随机微分方程，变换后的方程存在唯一解. 然后由变换函数的性质可得逆函数的存在性和Lipschitz连续性. 最后由Itô公式及逆函数的性质可得原来的McKean-Vlasov随机微分方程及相应的粒子系统解的存在唯一性.
- 高维McKean-Vlasov随机微分方程 /
- 粒子系统 /
- 逐段Lipschitz连续 /
- Zvonkin变换 /
- 解的存在唯一性
Abstract: The existence and uniqueness of the solutions to high-dimensional McKean-Vlasov stochastic differential equations with discontinuous drift coefficients and corresponding particle systems, were investigated. With the drift coefficient being piecewise Lipschitz continuous about the space variable, through Zvonkin's transformation, the original equation was converted into a new McKean-Vlasov stochastic differential equation with Lipschitz continuous coefficients. Therefore, the new equation has a unique solution. Moreover, the existence and Lipschitz continuity of the inverse function were proven according to the transformation function characteristics. Finally, based on the It's formula and the inverse function characteristics, the existence and uniqueness of the solutions to the McKean-Vlasov stochastic differential equation and the corresponding particle system were obtained.

HTML全文

参考文献(17)

[1]	LEOBACHER G, SZOLGYENYI M. A numerical method for SDEs with discontinuous drift[J]. BIT Numerical Mathematics, 2016, 56 (1): 151-162. doi: 10.1007/s10543-015-0549-x
[2]	LEOBACHER G, SZOLGYENYI M. A strong order 1/2 method for multidimensional SDE with discontinuous drift[J]. The Annals of Applied Probablity, 2015, 27 (4): 2383-2418.
[3]	马丽, 马瑞楠. 一类随机泛函微分方程带随机步长的EM逼近的渐近稳定[J]. 应用数学和力学, 2019, 40 (1): 97-107. doi: 10.21656/1000-0887.390057 MA Li, MA Ruinan. Almost sure asymptotic stability of Euler-Maruyama method with random variable stepsizes for stochastic functional differential equations[J]. Applied Mathematics and Mechanics, 2019, 40 (1): 97-107. (in Chinese) doi: 10.21656/1000-0887.390057
[4]	梁青. 一类带扰动的随机脉冲泛函微分方程解的渐近性[J]. 应用数学和力学, 2022, 43 (9): 1034-1044. doi: 10.21656/1000-0887.420267 LIANG Qing. Asymptotic properties of the solution to a class of perturbed stochastic impulsive functional differential equations[J]. Applied Mathematics and Mechanics, 2022, 43 (9): 1034-1044. (in Chinese) doi: 10.21656/1000-0887.420267
[5]	李光洁, 杨启贵. G-Brown运动驱动的非线性随机时滞微分方程的稳定化[J]. 应用数学和力学, 2021, 42 (8): 841-851. doi: 10.21656/1000-0887.410332 LI Guangjie, YANG Qigui. Stabilization of nonlinear stochastic delay differential equation driven by G-Brownian motion[J]. Applied Mathematics and Mechanics, 2021, 42 (8): 841-851. (in Chinese) doi: 10.21656/1000-0887.410332
[6]	SCHEUTZOW M. Uniqueness and non-uniqueness of solutions of Vlasov-McKean equations[J]. Journal of the Australian Mathematical Society, 1987, 43 (2): 246-256. doi: 10.1017/S1446788700029384
[7]	LEOBACHER G, REISINGER C, STOCKINGER W. Well-posedness and numerical schemes for one-dimensional McKean-Vlasov equations and interacting particle systems with discontinuous drift[J]. BIT Numerical Mathematics, 2022, 62 (4): 1505-1549. doi: 10.1007/s10543-022-00920-4
[8]	ROCKNER M, ZHANG X. Well-posedness of distribution dependent SDEs with singular drifts[J]. Bernoulli, 2021, 27 (2): 1131-1158.
[9]	HUANG X, WANG F. Singular McKean-Vlasov (reflecting) SDEs with distribution dependent noise[J]. Journal of Mathematical Analysis and Applications, 2020, 154 (1): 126301.
[10]	WANG F. Distribution dependent SDEs for Landau type equations[J]. Stochastic Processes and Their Applications, 2018, 128 (2): 595-621.
[11]	CHAUDRU DE RAYNAL P E. Strong well posedness of McKean-Vlasov stochastic differential equations with Hölder drift[J]. Stochastic Processes and Their Applications, 2020, 130 (1): 79-107.
[12]	REN P. Singular McKean-Vlasov SDEs: well-posedness, regularities and Wang's Harnack inequality[J]. Stochastic Processes and Their Applications, 2023, 156 : 291-311.
[13]	ZHAO G. On distribution dependent SDEs with singular drifts[R/OL].[2023-03-11]. https://arxiv.org/abs/2003.04829v3.
[14]	HAMMERSLEY W, SISKA D, SZPRUCH L. McKean-Vlasov SDEs under measure dependent Lyapunov conditions[J]. Annales de l'Institut Henri Poincare, Probabilites et Statistiques, 2021, 57 (2): 1032-1057.
[15]	CARMONA R, DELARUE F. Probabilistic Theory of Mean Field Games With Applications [M]. Springer Cham, 2018.
[16]	RUZHANSKY M, SUGIMOTO M. On global inversion of homogeneous maps[J]. Bulletin of Mathematical Sciences, 2015, 5 (1): 13-18.
[17]	MAO Xuerong. Stochastic Differential Equations and Applications[M]. Chichester: Horwood Publishing Ltd, 2008.